Über die starke Summierbarkeit von Orthogonalreihen durch Euler-Verfahren

, (2) (1), (E, q) (0<q<1), , , , m _i =i ² (. . . [], X. [4]). , (1) (E, q) (0<q<1) (E, q), . .      

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

Nonparametric Inference for a Class of Stochastic Partial Differential Equations II

Consider the stochastic partial differential equationdu (t,x) = (t)u (t, x)dt + dW _Q(t,x), 0 t T where = ₂/x ₂, and is a class of positive valued functions. We obtain an estimator for the linear multiplier (t) and establish the consistency, rate of convergence and asymptotic normality of this estimator as 0.     

A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.     

A proof for Theodorus' theorem by drawing diagrams

Summary In Plato's dialogue Theaetetus, the mathematician Theodorus (ca. 470-400 BC) is said to have proved the irrationality of 3, 5, ..., 17 by drawing diagrams. In this paper such a proof is presented and compared with another, more arithmetical, reconstructed proof by W. Knorr. Moreover a hypothetical early theory of similar rectangles is extracted from Euclid's Elements, which could have served as a basis for the geometrical proof of Theodorus' theorem.Herrn N. Stephanidis in Freundschaft gewidmet     

Die Methode der „direkten Koeffizientenanpassung“ (μ-Form) des Separable Programming

Zusammenfassung Zur approximativen Lösung von Optimierungsproblemen mit trennbaren nichtlinearen Funktionen (Separable Programming) werden häufig modifizierte Methoden der linearen Planungsrechnung verwendet. Dabei ersetzt man die nichtlinearen Funktionen durch lineare Streckenzüge. Für diese polygonale Approximation werden in der Literatur die sog.-Form und die sog.-Form vorgeschlagen, die jedoch eine aufwendige Datenorganisation erfordern. In diesem Beitrag soll eine kompaktere Organisationsform (-Form) vorgeschlagen werden. Zu ihrer Durchführung werden die Rechenregeln der Upper-Bounding-Technique und die Regeln zur nachträglichen Änderung von Matrixelementen verwendet. Der Vorteil gegenüber der-Form und der-Form liegt in einem wesentlich geringeren Speicherplatzbedarf, im niedrigeren Rechenaufwand und in einer größeren Flexibilität.

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Summary Separable programming problems can be solved by modifications of the simplex method, when the single functions are replaced by linear polygonal approximations. Two forms of different data organization are discussed in the literature: the-form and the-form. In this paper a more compact data organization (-form) is suggested. It uses the upper bounding technique and the formulas for the subsequent alteration of matrix elements. The advantage of this method over both the-form and the-form is due to much less storage space, to a reduced computation time, and to a higher flexibility.

     

Famiglie di dischi analitici con bordo su sottovarietàCR non generiche

Summary In this work the authors prove that all sufficiently small analytic discs in the tangent space of a non generic embedded CR manifold MC^N (M at least of class C⁴) can be lifted uniquely to analytic discs in C^N with boundaries on M Moreover if M is real analytic or C then real analytic or C discs lift without loss of derivatives. If M is of class C^K then there is a 1+ loss of derivatives in the lifting. A stability result is also proven.     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

On the absolute Riesz summability of orthogonal series

- , [4]. , .     

Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy

Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) u_t=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦²/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u).     

On the exactness of a theorem of G.A. Fomin

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Асимптоический Анализ Распределения Статистики Диксона

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Compound Poisson Processes with Two-Sided Reflection

We consider a compound oscillating Poisson process with two-sided reflection. This process is defined by an upper-semicontinuous compound Poisson process (t) and its functionals, namely the first-exit time of (t) from an interval and the first-exit time of (t) across the upper and lower levels. We study the main characteristics of this oscillating process in terms of the potential and resolvent of the process (t) introduced by Korolyuk. For this purpose, we refine the Pecherskii identities and some other results for upper-semicontinuous Poisson processes.     

Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

. - . .. . . . , 340 048 . 1      

On the admissibility of function spaces of typeB p,q s andF p,q s,and boundary value problems for non-linear partial differential equations

u=f(x)+S(u), S — , u-G(u), G — . B _p,q ^s () -F _p,q ^s (). R _n — . — . _p,q ^s F _p,q ^s .     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Topological and Nonstandard Extensions

We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the Stone-ech compactification X of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a natural topology). In this context, we give a simple characterization of nonstandard extensions in purely topological terms, and we establish connections with special classes of ultrafilters whose existence is independent of ZFC.     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Role of information in the stochastic zero-sum differential game

In this paper, a linear-quadratic Gaussian zero-sum differential game is studied. Maneuverability is defined to measure players' strength. It is shown that a more maneuverable player would prefer a more observable information system. An example is given to show that a more controllable player might not prefer more observable measurements in the stochastic environment.The research reported in this paper was made possible through support extended to the Division of Engineering and Applied Physics, Harvard University, by the US Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-c-0648 and by the National Science Foundation under Grant No. GK31511.     

Vacuum polarization of a complex automorphic scalar field in two-dimensional spacetime with closed null geodesics and the time-machine problem

A quantized automorphic scalar field in two-dimensional spacetime with closed null geodesics (Cauchy horizon) is considered. The renormalized energy—momentum tensor T _v^ren is obtained. It is shown that for specific values of the automorphic parameter T _v^ren remains regular on the Cauchy horizon.Kazan Pedagogical Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 1 pp. 134–149, January, 1995.